Most mathematics teachers encountered the Method of Exhaustion first when they studied the definite integral in calculus. However, the Method of Exhaustion predates and anticipates the development of calculus by many centuries, and the discussion of the computation of the area of a circle in Section 10.1.4 illustrates how the Method of Exhaustion was used long before the invention of calculus. In that discussion, the given circle C of radius r is inscribed with regular polygons s[n] and circumscribed with regular polygons S[n] with 2 to the power n sides in such a way that:
The area of the circle C is the unique positive real number that is larger than s[n] and smaller than S[n] for all positive integers n. Thus, the area of the circle is “exhausted” by the increasing areas of the inscribed regular polygons and the decreasing areas of the circumscribed regular polygons.
The application of the Method of Exhaustion to the definition and calculation of the definite integral in calculus is technically simpler than the preceding application to defining and computing the area of a circle as we will now see.
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One of the first applications of the definite integral that encountered in your calculus course is to the calculation of the area bounded by the graphs of two functions f and g over an interval [a, b] of the x-axis. For example, the graph below shows an area bounded by the graphs of two functions f and g over the interval [-3, 2] of the x-axis:
This area is computed with the following definite integral:
While this calculation is a routine application of calculus, it does use three of the most powerful tools in calculus: the definite integral, the anti-derivative, and The Fundamental Theorem Of Calculus. It is remarkable that Archimedes had developed the tools necessary to compute the areas of “parabolic sections” such as that given above long before calculus was invented. His tools were based on an ingenious application of the Method of Exhaustion and his discovery of a remarkable property of parabolas. As a high school math teacher, you may think that there is not much about parabolas that you have not seen before but I think that you will find the following information to be new and very interesting. At least I did!
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